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Almost Periodic Solutions of a Class of Singularly Perturbed Differential

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Almost Periodic Solutions of a Class of Singularly Perturbed Differential http://www.paper.edu.cn Nonlinear Analysis 37 (1999) 841–859 Almost periodic solutions of a class of singularly perturbed di erential equations with piecewise constant argument 1 Rong Yuan Department of Mathematics, Beijing Normal University, Beijing 100875, People’s Republic of China Received 27 March 1997; accepted 13 October 1997 Keywords: Almost periodic solutions; Almost periodic sequences; Piecewise constant argument; Singular perturbation 1. Introduction The main purpose of this paper is to show the existence of almost periodic solutions to the following singularly perturbed systems of di erential equations with piecewise constant argument ( 0 N N x (t)=F(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ;); (1.1) 0 N N y (t)=G(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ;) in the case that F and G are almost periodic for t uniformly on R2N+2 × R2N+2, where ¿0 is a small parameter, x∈R; y∈R, and [·] denotes the greatest integer function. There have been many celebrated works ([1,3,5–7,13,14,16,18–20] and the references cited therein) concerning with di erential equations with piecewise constant argument. The ÿrst contribution on these equations is due to Cooke and Wiener [5], and Shah and Wiener [16]. It seems to us that the strong interest in di erential equations with piecewise constant argument is motivated that they describe hybrid dynamical sys- tems (a combination of continuous and discrete). In their comprehensive survey paper, Cooke and Wiener [6] describe recent progress in the area of di erential equations with piecewise constant argument, from which we know that all of the works that has been done on the di erential equations concerns the stability, the oscillation and the 1 Supported by the National Natural Science Foundation of China. 0362-546X/99/$ – see front matter ? 1999 Elsevier Science Ltd. All rights reserved. 转载 PII: S0362-546X(98)00076-5 中国科技论文在线 http://www.paper.edu.cn 842 R. Yuan / Nonlinear Analysis 37 (1999) 841–859 existence of periodic solutions, and none of the works concerns the existence of almost periodic solutions. In [13, 14], Papaschinopoulos studied the asymptotic behavior for these equations. As is well known, the existence problem of periodic solutions and almost periodic solutions has been one of the most attracting topics in the qualitative theory of ordinary or functional di erential equations for its signiÿcance in the physical sciences. There have been many remarkable works ([4, 8, 10–12, 20] and the references cited therein) concerning the existence of almost periodic solutions. For a special form of singularly perturbed di erential equation (1.1): ( x0(t)=F(t; x(t);y(t);); (1.2) y0(t)=G(t; x(t);y(t);); the existence of periodic solutions was ÿrst studied by Flatto and Levinson [9]. Hale and Seifert [10] and Chang [4] generalized Flatto and Levinson’s results and investigated the existence of almost periodic solutions to Eq. (1.2). Recently, Smith [17] also showed the existence and stability of almost periodic solutions to Eq. (1.2). The motivation of this paper comes from Smith’s paper and the author’s paper [20]. The present paper will concentrate on the study of the existence of almost periodic solutions to Eq. (1.1) of neutral type (that is, we consider equations with argument [t]; [t − n]; [t + n], where n is a natural number). To our knowledge, only paper [20] was concerned with the existence of almost periodic solutions to di erential equations with piecewise constant argument in all published papers. Clearly, Eq. (1.1) is a generalization form of Eq. (1.2). It is assumed that the degenerate system ( 0 N N x (t)=F(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ; 0); (1.3) N N 0=G(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ; 0) has an almost periodic “outer” solution which we take to be the trivial solution, that is, we suppose F(t; 0;:::;0; 0) ≡ G(t; 0;:::;0; 0) ≡ 0 so that (x; y)=(0; 0) satisÿes Eq. (1.3). Our aim is to seek for almost periodic so- lutions of Eq. (1.1) near the “outer” solution. Expanding Eq. (1.1) about the trivial solution gives N N X X x0(t)=a(t; )x(t)+ a (t; )x([t + i]) + b(t; )y(t)+ b (t; )y([t + i]) i i i=−N i=−N N N + f(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ;); N N X X y0(t)=c(t; )x(t)+ c (t; )x([t + i]) + d(t; )y(t)+ d (t; )y([t + i]) i i i=−N i=−N N N + g(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ;): (1.4) 中国科技论文在线 http://www.paper.edu.cn R. Yuan / Nonlinear Analysis 37 (1999) 841–859 843 One can think of, e.g., a(t; )as@F=@x(t; 0;:::;0;), but, in fact, it is really (1.4) which we study in this paper. In what follows, we denote by |·| the Euclidean norm and by [·] the greatest integer function. We say that a function (x; y):R → R × R is a solution of Eq. (1.4) (or Eq. (1.1)) if the following conditions are satisÿed (i) (x; y) is continuous on R, (ii) the derivative (x0;y0)of(x; y) exists on R except possibly at the point t = n; n∈ Z = {:::;−1; 0; 1;:::} where one-sided derivative exists, (iii) (x; y) satisÿes Eq. (1.4) (or Eq. (1.1)) in the intervals (n; n +1);n∈Z. The following hypotheses are assumed to hold throughout the paper. (H1) a(t; );ai(t; );b(t; );bi(t; );c(t; );ci(t; );d(t; );di(t; );i=0; 1;:::;N, are almost periodic functions in t. They are continuous in , uniformly in t ∈R. Let M denote a common bound for these functions on (t; )∈R × [0;0]. 0 0 0 0 (H2) a(t; 0) = a ;ai(t; 0) = ai ;c(t; 0)=0;ci(t; 0)=0;d(t; 0) = d ;di(t; 0) = di ;i=0; 1;:::;N, are constants and d0¡0. (H3) All roots 1;:::;2N of algebraic equation XN i Ai =0 i=−N are simple and |i|6=1; 1 ≤ i ≤ 2N, where a0 0−1 0 a0 0−1 0 a0 A0 =e + a a0(e − 1);A1 = a a1(e − 1) − 1; 0−1 0 a0 Ai = a ai (e − 1);i= −1; 2;:::;N: (H4) All roots 1;:::;2N of algebraic equation XN i Di = 0 (1.5) i=−N are simple and |i|6=1; 1 ≤ i ≤ 2N, where 0−1 0 0−1 0 D0 = −d d0;D1 = −d d1 − 1; 0−1 0 Di = −d di ;i= −1; 2;:::;N: (H5) f; g are almost periodic in t uniformly on (x0;:::;x2N+1;y0;:::;y2N+1) such that t ∈R; |xi|; |yi|≤0; (i =0;:::;2N +1); 0 ≤ ≤ 0; 0 ≤ ≤ 0. Furthermore, there exist nondecreasing functions M() and Á(; ); 0 ≤ ≤ 0; 0 ≤ ≤ 0 sat- isfying lim→0 M()=0; lim(; )→(0; 0) Á(; ) = 0, such that |f(t; 0;:::;0;)|≤M(); |g(t; 0;:::;0;)|≤M();t∈R; 0 ≤ ≤ 0 中国科技论文在线 http://www.paper.edu.cn 844 R. Yuan / Nonlinear Analysis 37 (1999) 841–859 and |f(t; x0;:::;x2N+1;y0;:::;y2N+1;) − f(t; x0;:::;x2N+1; y0;:::;y2N+1;)| 2XN+1 ≤ Á(; ) [|xi − xi| + |yi − yi|]; i=0 |g(t; x0;:::;x2N+1;y0;:::;y2N+1;) − g(t; x0;:::;x2N+1; y0;:::;y2N+1;)| 2XN+1 ≤ Á(; ) [|xi − xi| + |yi − yi|] i=0 hold for all t ∈R; |xi|; |xi|; |yi|; |yi|≤0; (i =0;:::;2N +1); 0 ≤ ≤ 0. We are now in a position to formulate our main theorem. Main Theorem. If (H1)–(H5) hold, then there exist 2;0; 0¡2 ≤ 0 and 0¡1 ≤ 0; such that for each satisfying 0¡ ≤ 2, (1.4) has a unique almost periodic solution ∗ ∗ (x (t; );y (t; )) satisfying kxk≤1; kyk≤1 and this solution is continuous in uniformly in t ∈R and satisÿes kx∗()k + ky∗()k = O(M()) as → 0. Furthermore, if f; g are !-periodic in t; then the following results hold: + ∗ ∗ (1) If ! = n0 ∈Z ; then the unique solution (x (t; );y (t; )) is !-periodic in t. + (2) If ! = n0=m0;n0;m0 ∈Z ;n0 and m0 are mutually prime, then the unique solution ∗ ∗ (x (t; );y (t; )) is m0!-periodic in t. This paper is organized as follows. In Section 2, we list some well-known results and investigate the existence of almost periodic solutions to EPCA with a parameter by using the idea in [20]. The proof of main theorem will be given in Section 3. 2. Some lemmas For a beginning we introduce the deÿnition of almost periodic functions. Deÿnition 1 (Fink [8]). A function f : R → Rp is called an almost periodic function if the Á-translation set of f T(f; Á)={∈R ||f(t + ) − f(t)|¡Á; ∀t ∈R} is a relatively dense set in R for all Á¿0. is called the Á-period for f. Deÿnition 2 (Fink [8] and Meisters [12]). A sequence x : Z → Rp is called an almost periodic sequence if the Á-translation set of x T(x; Á)={∈Z ||x(n + ) − x(n)|¡Á; ∀n∈Z} 中国科技论文在线 http://www.paper.edu.cn R. Yuan / Nonlinear Analysis 37 (1999) 841–859 845 is a relatively dense set in Z for all Á¿0.
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